Advertisements
Advertisements
प्रश्न
A triangle is formed by joining the points (1, 0, 0), (0, 1, 0) and (0, 0, 1). Find the direction cosines of the medians
Advertisements
उत्तर
`vec"OA" = hat"i", vec"OB" = hat"j", vec"OC" = hat"k"`
D is the midpoit of BC
∴ `vec"OD" = (vec"OB" + "OC")/2`
= `(hat"j" + hat"k")/2`
E is the midpoint of AC
`vec"OE" = (hat"i" + hat"k")/2`
F is the midpoint of AB
∴ `vec"OF" = (hat"i" + hat"j")/2`
Now the medians are `vec"AD", vec"BE"` and `vec"CF"`
(i) `vec"AD" =vec"OD" - vec"OA" = (hat"j" +hat"k")/2 - hat"i"`
= `-hat"i" + hat"j"/2 + hat"k"/2`
`|vec"AD"| = sqrt(1 + 1/4 + 1/4)`
= `sqrt(1 + 1/2)`
= `sqrt(3)/sqrt(2)`
and d.c's of `vec"AD" = ((-1)/(sqrt(3)/sqrt(2)), (1/2)/(sqrt(3)/sqrt(2)), (1/2)/(sqrt(3)/sqrt(2)))`
= `(- sqrt(2)/sqrt(3), sqrt(2)/(2sqrt(3)), sqrt(2)/(2sqrt(3)))`
= `(- sqrt(2)/sqrt(3), 1/(sqrt(2)sqrt(3)), 1/(sqrt(2)/sqrt(3)))`
= `(- sqrt(2)/sqrt(3), 1/sqrt(6), 1/sqrt(6))`
`["Now" sqrt(2)/sqrt(3) = sqrt(2)/sqrt(3) xx sqrt(2)/sqrt(2) = 2/sqrt(6)] = (- 2/sqrt(6), 1/sqrt(6), 1/sqrt(6))`
(ii) `vec"BE" = vec"OE" - vec"OB"`
= `(hat"i" + hat"k")/2 -hat"j"`
= `hat"i"/2 - hat"j" + hat"k"/2`
`|vec"BE"| = sqrt(1/4 + 1 + 1/4)`
= `sqrt(3)/sqrt(2)`
d.c's of `vec"BE" = ((1/2)/(sqrt(3)/sqrt(2)), (-1)/(sqrt(3)/sqrt(2)), (1/2)/(sqrt(3)/sqrt(2))) = (1/sqrt(6), (-2)/sqrt(6), 1/sqrt(6))`
(iii) `vec"CE" = vec"OF" - vec"OC"`
= `(hat"i" + hat"j")/2 - hat"k"`
= `hat"i"/2 + hat"j"/2 - hat"k"`
`|vec"CF"| = sqrt(1/4 + 1/4 + 1)`
= `sqrt(3)/sqrt(2)`
d.c's of `vec"CF" = ((1/2)/(sqrt(3)/sqrt(2)), (1/2)/(sqrt(3)/sqrt(2)), (-1)/(sqrt(3)/sqrt(2)))`
= `(1/sqrt(6), 1/sqrt(6), (-2)/sqrt(6))`
APPEARS IN
संबंधित प्रश्न
If l, m, n are the direction cosines of a line, then prove that l2 + m2 + n2 = 1. Hence find the
direction angle of the line with the X axis which makes direction angles of 135° and 45° with Y and Z axes respectively.
Find the angle between the vectors with direction ratios proportional to 1, −2, 1 and 4, 3, 2.
Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.
Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the line determined by the points (3, 5, −1) and (4, 3, −1).
If the coordinates of the points A, B, C, D are (1, 2, 3), (4, 5, 7), (−4, 3, −6) and (2, 9, 2), then find the angle between AB and CD.
Write the ratio in which YZ-plane divides the segment joining P (−2, 5, 9) and Q (3, −2, 4).
If a unit vector `vec a` makes an angle \[\frac{\pi}{3} \text{ with } \hat{i} , \frac{\pi}{4} \text{ with } \hat{j}\] and an acute angle θ with \[\hat{ k} \] ,then find the value of θ.
For every point P (x, y, z) on the x-axis (except the origin),
If P (3, 2, −4), Q (5, 4, −6) and R (9, 8, −10) are collinear, then R divides PQ in the ratio
The direction ratios of the line which is perpendicular to the lines with direction ratios –1, 2, 2 and 0, 2, 1 are _______.
Find the direction cosines and direction ratios for the following vector
`3hat"i" - 4hat"j" + 8hat"k"`
Find the direction cosines and direction ratios for the following vector
`3hat"i" - 3hat"k" + 4hat"j"`
Find the direction cosines and direction ratios for the following vector
`hat"i" - hat"k"`
If `vec"a" = 2hat"i" + 3hat"j" - 4hat"k", vec"b" = 3hat"i" - 4hat"j" - 5hat"k"`, and `vec"c" = -3hat"i" + 2hat"j" + 3hat"k"`, find the magnitude and direction cosines of `3vec"a"- 2vec"b"+ 5vec"c"`
The x-coordinate of a point on the line joining the points Q(2, 2, 1) and R(5, 1, –2) is 4. Find its z-coordinate.
A line makes equal angles with co-ordinate axis. Direction cosines of this line are ______.
O is the origin and A is (a, b, c). Find the direction cosines of the line OA and the equation of plane through A at right angle to OA.
The direction cosines of vector `(2hat"i" + 2hat"j" - hat"k")` are ______.
The Cartesian equation of a line AB is: `(2x - 1)/2 = (y + 2)/2 = (z - 3)/3`. Find the direction cosines of a line parallel to line AB.
If a line makes angles of 90°, 135° and 45° with the x, y and z axes respectively, then its direction cosines are ______.
